#461461

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,34)(21,24)(22,28)(23,27)(25,32)(26,37)(30,43)(31,42)(33,39)(35,41)(36,44)(38,40)(45,46)(47,48);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,24)(25,39)(26,40)(27,28)(29,42)(34,44)(41,43)(45,48);(1,3)(2,6)(4,7)(5,8)(11,15)(13,16)(17,19,23,33,21,31,22,32)(18,25,28,42,24,39,27,29)(20,35,38,46,36,30,37,47)(26,43,44,45,40,41,34,48);(3,7)(6,8)(17,20)(18,26)(19,30)(21,36)(22,37)(23,38)(24,40)(25,41)(27,34)(28,44)(29,45)(31,35)(32,46)(33,47)(39,43)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368657);